Direct and inverse proportion: ratio or product?

1. Direct and inverse proportion: name the quantities

A quantity includes a meaning and a unit. In a printing problem, write number of identical sheets and total printing cost, rather than two unexplained letters. In a journey problem, write average speed in kilometres per hour and travel time in hours. Decide which value is known and which one the question asks you to find. A tidy two-column table keeps corresponding values together; crossing values from different situations can produce a plausible-looking but meaningless calculation.

Then record what is held fixed. Printing cost may be directly proportional to sheet count only when the price per sheet stays unchanged and there is no initial fee. Speed and travel time may be inversely proportional when the distance stays unchanged. The same speed can instead be directly proportional to distance when travel time is fixed. The story's subject does not determine the relationship. The quantities you compare and the conditions you hold constant determine it.

Try a simple prediction before using numbers. If twice as many sheets are printed at an unchanged unit price, the cost should double. If average speed doubles over the same distance, time should halve. These predictions help detect a reversed calculation later. They are conditional statements, not universal rules about printers or travel. For a fuller introduction to the algebraic notation, consult OpenStax on direct and inverse variation; the examples in this guide are our own.

2. Test a table with ratios, not just increases

Suppose an exercise gives these paired values: 2 sheets cost 6 units of money, 5 sheets cost 15, and 8 sheets cost 24. Divide cost by sheet count in every row. The results are 6/2 = 3, 15/5 = 3 and 24/8 = 3. That constant ratio supports the direct-proportion model y = 3x for the supplied data. The constant has a meaning: three money units per sheet. Write that meaning beside the calculation instead of leaving a bare three.

Under this model, 11 sheets cost 33 units. Multiplying the sheet count by 11/5 also multiplies the cost by 11/5; you do not need an integer scaling factor. An alternative is to find the cost of one sheet and multiply by eleven. Both methods express the same relationship. Use whichever you can explain clearly. When a task gives a complete table, check every supplied pair, not merely the pair closest to the missing entry.

Now change the costs to 10, 19 and 28 for sheet counts 2, 5 and 8. Both columns still increase, but the ratios are 5, 3.8 and 3.5. The relationship is not directly proportional. It fits a fixed charge of four plus three per sheet: y = 3x + 4. Recognising this counterexample matters more than memorising the phrase more means more. Constant differences can indicate a linear relationship, but a nonzero starting value prevents direct proportionality.

3. Test inverse proportion with a constant product

Imagine rectangular display panels that must each have an area of 48 square centimetres. Their possible width and height pairs are 2 and 24, 4 and 12, 6 and 8, and 8 and 6 centimetres. Multiply each pair. Every product equals 48. Width and height are inversely proportional under the fixed-area condition, so height = 48/width. Doubling width halves height; multiplying width by three divides height by three. The constant product represents area, not a height or a width.

For a width of 12 centimetres, the height is 48/12 = 4 centimetres. Check the result by multiplication: 12 × 4 = 48 square centimetres. A mistaken direct calculation might produce a taller panel as width increases. The original fixed-area condition immediately exposes that problem. In this positive-length model, neither width nor height can be zero. Division by zero is undefined, and a zero-width panel cannot have the stated positive area.

A decreasing table alone is insufficient. Pairs 2 and 18, 4 and 16, 6 and 14 all move in opposite directions, but their products are 36, 64 and 84. They follow y = 20 − x, not y = k/x. Keep a third possible verdict on your page: neither. If your arithmetic foundations need attention, the Lirno mathematics study guide discusses explaining steps; you can still carry out the constant-product test independently on paper.

4. Solve a missing value after choosing the model

For a direct example, 7 identical notebooks cost 21 money units with no discount or fixed charge. The unit cost is 21/7 = 3, so 12 notebooks cost 12 × 3 = 36. An equation reaches the same answer: 21/7 = c/12. Cross-multiplication is valid because you have already established equal ratios. It does not decide whether the situation is direct. Treating a familiar four-number layout as permission to cross-multiply skips the important reasoning step.

For an inverse example, a journey of fixed length takes 3 hours at an average speed of 40 kilometres per hour. The fixed distance is 40 × 3 = 120 kilometres. At 60 kilometres per hour, time is 120/60 = 2 hours. Equivalently, 40 × 3 = 60 × t. Time decreases by the reciprocal of the speed factor: speed increases by 1.5, so time is divided by 1.5. This uses average speed over the complete journey, not a speed observed at one instant.

Write the model, substitute the known pair, find the constant, calculate the missing value and check against the original condition. That sequence is easier to audit than arrows with no labels. You can draw arrows as reminders, but include the factor on each arrow and explain why the second factor is the same or reciprocal. If the prompt does not establish proportionality or enough information to justify it, say what additional assumption you need instead of manufacturing a unique answer.

5. Keep units and realistic conditions visible

The inverse journey example works only when speed and time use compatible units. If the given time is 90 minutes, convert it to 1.5 hours before multiplying by kilometres per hour. At 48 kilometres per hour, the distance is 72 kilometres, not 4,320 kilometres. The latter number results from treating minutes as hours. Label each quantity through the calculation; cancellation of hours helps you see why the remaining unit is kilometres. Convert deliberately before choosing the numerical operation.

Work-rate problems need particularly careful assumptions. Four identical machines taking six hours to complete a fixed batch suggests 24 machine-hours. Six such machines would take four hours if they work at the same constant rate, operate together and encounter no shared bottleneck. Real machines can have setup time or limited material supply. Groups of people also vary in productivity and may spend time coordinating. In a school model, state the specified equal-rate assumption; do not present it as a prediction guaranteed in real life.

6. Use a graph as a check, not a shortcut

For positive quantities in direct proportion, y = kx produces a straight line through the origin. In the sheets example, points such as (2, 6), (5, 15) and (8, 24) lie on that line. A straight line that crosses the vertical axis above zero instead includes an initial amount and is not direct proportion. Check the labelled axes and their units before interpreting steepness. A different plotting scale can make the same mathematical line look much steeper on the page.

For positive inverse proportion, y = k/x produces a descending curve rather than a straight line. The fixed-area pairs approach the axes as width or height grows, but the model does not reach either axis at a finite positive value. Join points with the curve appropriate to the model when the assignment asks for it, rather than inventing straight sections between sparse points. For practical graph construction, the guide to plotting axes and scale explains how to keep coordinate choices readable.

7. Explain the relationship in a complete sentence

A strong answer links a calculation to its meaning. For direct proportion, write: the cost divided by the number of sheets is three in every supplied row, so the cost is directly proportional to sheet count under the unchanged-price condition. For inverse proportion, write: each width multiplied by its corresponding height is 48, so the dimensions are inversely proportional when area is fixed. These explanations identify the quantities, the constant and the condition that makes the model usable.

For neither, give a counterexample rather than a vague rejection. In the fixed-charge table, cost per sheet changes from five to 3.8, which rules out a constant ratio. If someone claims inverse proportion, compare the products too. You do not need to calculate every possible future value to disprove an exact rule: one conflicting pair is enough. Checking all supplied rows still helps identify whether the issue is a model choice, a copied value or arithmetic.

If you use Lirno to clarify one uncertain step, keep your own attempt and ask a narrow question about the ratio or product. AI can misread values or reason incorrectly; compare its explanation with the original table and your independent calculation. School rules still govern permitted help, and Lirno cannot guarantee correctness, grades, mastery or permission. The guide to using AI without copying discusses those boundaries. A tool should help you articulate the model, not replace your evidence.

8. Practise choosing between three possible answers

Try three short datasets before calculating a missing value. First, x values 3, 6 and 9 correspond to y values 12, 24 and 36. Second, x values 3, 6 and 9 correspond to y values 12, 6 and 4. Third, x values 3, 6 and 9 correspond to y values 8, 11 and 14. Write direct, inverse or neither, and support each choice with numbers. Do not rely on which column grows. In the third set, consider whether an initial amount explains the pattern.

The first set has y/x = 4 throughout, so y = 4x and x = 12 gives y = 48. The second has xy = 36 throughout, so y = 36/x and x = 12 gives y = 3. The third fits y = x + 5, but has neither a constant ratio nor a constant product. It is a useful reminder that a predictable relationship can still fall outside both proportional models. These are original practice examples, not predictions about a real process.

For a final self-check, change one detail in a story: add a starting fee, change the fixed distance or allow unequal machine rates. Explain which earlier calculation stops being justified and why. Finish by checking the model, units, magnitude and original condition separately. When you can describe what stays constant and recognise when nothing supplied stays constant, a missing-value problem becomes a reasoned calculation rather than a guess about multiplying or dividing.